Properties of $n$-ary groups connected with the affine geometry are considered. Some conditions for an $n$-ary $rs$-group to be derived from a binary group are given. Necessary and sufficient conditions for an $n$-ary group $<\theta ,b>$-derived from an additive group of a field to be an $rs$-group are obtained. The existence of non-commutative $n$-ary $rs$-groups which are not derived from any group of arity $m<n$ for every $n\ge 3$, $r>2$ is proved.

Bz the quadrileteral condition in a given net there is meant the following implication: If $A_1,A_2,A_3,A-4$ are arbitrary points, no three of them lie on the same line, with coll $\left(A_i{A}_j\right)$ (collinearity) for any five from six couples $\{i,j\}$ then there follows the collinearity coll $\left(A_k{A}_l\right)$ for the remaining couple $\{k,l\}$. In the article there is proved the every net satisfying the preceding configuration condition is necessarity the Ostrom net (i.e., the net over a field). Conversely, every Ostrom net satisfies the above configuration...